For a simple graph G=(V,E) with no isolated vertices, total Roman {3}-dominating function(TR3DF) on G is function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 f(v)=1; and (iii) every vertex v f(v)≠0 has neighbor u f(u)≠0 for v∈V(G). The weight of TR3DF f sum f(V)=∑v∈V(G)f(v) minimum called {3}-domination number denoted by γt{R3}(G). In this paper, we...

Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the pro...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

This paper starts from four observations: (1) voter turnout is declining in established democracies; (2) low turnout means socio-economically unequal turnout; (3) compulsory voting is an effective means to increase turnout; (4) even low-turnout countries, however, have neither introduced nor even contemplated a legal obligation to vote. A closer look at the arguments against compulsory voting s...

Egypt, located on the isthmus of Africa, is an ideal region to study historical population dynamics due to its geographic location and documented interactions with ancient civilizations in Africa, Asia and Europe. Particularly, in the first millennium BCE Egypt endured foreign domination leading to growing numbers of foreigners living within its borders possibly contributing genetically to the ...

the aim of this article is studying the second war of a set of three wars of shapur i against romans empire by virtue of the text of the inscription kaabe-ie zartosht. in this article by study and analysis of different presented ideas about the year of the occurrence of second war of shapur i against romans and also roman emperor who fought with him whose name has not been mentioned in inscript...